orces and Vectors Notation: = name for force vectors, as is A, B, C, T and P = force component in the direction = force component in the direction h = cable sag height L = span length = name for resultant vectors = resultant component in the direction = resultant component in the direction tail = start of a vector (without arrowhead) tip = direction end of a vector (with arrowhead) T = name for a tension force = ais direction = ais direction W = name for force due to weight = angle = angle, in a trig equation, e. sin, that is measured between the ais and tail of a vector orce Characteristics orces have a point of application - size units of lb, K, N, kn direction to a reference sstem, sense - indicated b an arrow, or b sign convention (+/-) Classifications include: Static & Dnamic Structural tpes separated primaril into Dead Load and Live Load with further identification as wind, earthquake (seismic), impact, etc. igid Bod Ideal material that doesn t deform orces on rigid bodies can be internal - within or at connections or eternal - applied igid bodies can translate (move in a straight line) or rotate (change angle) Weight of truck is eternal (gravit) Push b driver is eternal eaction of the ground on wheels is eternal If the truck moves forward: it translates 1
If the truck gets put up on a jack: it rotates Transmissibilit: We can replace a force at a point on a bod b that force on another point on the bod along the line of action of the force. or the truck: Eternal conditions haven t changed = The same eternal forces will result in the same conditions for motion Transmissibilit applies to EXTENAL forces. INTENAL forces respond differentl when an eternal force is moved. DEINITION: 2D Structure - A structure that is flat and ma contain a plane of smmetr. All forces on this structure are in the same plane as the structure. Internal and Eternal orces Internal forces occur within a member or between bodies within a sstem Eternal forces represent the action of other bodies or gravit on the rigid bod orce Sstem Tpes Collinear all forces along the same line 2
Coplanar all forces in the same plane Space out there urther classification as Concurrent all forces go through the same point Parallel all forces are parallel Static Equilibrium Equilibrium eists when the force sstem on a bod or object produces no rotation or translation. Graphical Addition of orces and esultants Parallelogram law: when adding two vectors acting at a point, the result is the diagonal of the parallelogram The tip-to-tail method is another graphical wa to add vectors. P P With 3 (three) or more vectors, successive application of the parallelogram law will find the resultant O drawing all the vectors tip-to-tail in an order will find the resultant. ectangular orce Components and Addition It is convenient to resolve forces into perpendicular components (at 90). Parallelogram law results in a rectangle. Triangle rule results in a right triangle. 3
is: between & = = cos sin magnitudes are scalar and can be negative & are vectors in and direction = 2 2 tan = When 90 < < 270, is negative When 180 < < 360, is negative When 0 < < 90 and 180 < < 270, tan is positive When 90 < < 180 and 270 < < 360, tan is negative Addition (analticall) can be done b adding all the components for a resultant component and adding all the components for a resultant component., and 2 2 tan CAUTION: An interior angle,, between a vector and either coordinate ais can be used in the trig functions. BUT No sign will be provided b the trig function, which means ou must give a sign and determine if the component is in the or direction. or eample, sin opposite side, which whould be negative in! 4
Eample 1 (page 18) Steps: 1. GIVEN: Write down what s given (drawing and numbers). 2. IND: Write down what ou need to find. (resultant graphicall) 3. SOLUTION: 4. Draw the 40 lb and 90 lb forces to scale with tails at 0. (If the scale isn t given, ou must choose one that fits on our paper; ie. 1 inch = 30 lb.) 5. Draw parallel reference lines at the ends of the vectors. 6. Draw a line from O to the intersection of the reference lines 7. Measure the length of the line 8. Convert the line length b the scale into pounds (b multipling b the force measure and dividing b the scale value, i.e. X inches * 30 lb / 1 inch). 5
Alternate solution: 4. Draw one vector 5. Draw the other vector at the TIP of the first one (awa from the tip). 6. Draw a line from 0 to the tip of the final vector and continue at step 7 Equilibrant The force equal and opposite to a resultant, that allows a sstem to be in equilibrium, is called an equilibrant. Eample 2 (pg 22) P 3 = 20 N, P 4 = 17 N, 10 mm = 5 N. P 1 = 12 N, P 2 = 17 N, 150 60 105 45 17 N = = 12 N = 20 N = 12 N 17 N = = 17N 38 = 18 N = 17 N 38 18 N = = 20 N 6
Eample 3 (pg 18) Determine the resultant vector analticall with the component method. Cable Structures Cables have the same tension all along the length if the are not cut. The force magnitude is the same everwhere in the cable even if it changes angles. Cables CANNOT be in compression. (The fle instead.) High-strength steel is the most common material used for cable structures because it has a high strength to weight ratio. Cables must be supported b vertical supports or towers and must be anchored at the ends. leing or unwanted movement should be resisted. (emember the Tacoma Narrows Bridge?) Cables with a single load have a concurrent force sstem. It will onl be in equilibrium if the cable is smmetric. The forces anwhere in a straight segment can be resolved into and components of T T cos and T T sin. The shape of a cable having a uniform distributed load is almost parabolic, which means the geometr and cable length can be found with: 2 2 4h( L ) / L where is the vertical distance from the straight line from cable start to end h is the vertical sag (maimum ) is the distance from one end to the location of L is the horizontal span. L h 7
2 L 32 4 h total 5 4 L( 1 8 h 3 2 L L ) where L total is the total length of parabolic cable h and L are defined above. Eample 4 Using force polgons and component relationships, determine the magnitudes in cables BC and CA. 8